Algebraic toplogy II
Math 540b, USC Spring 2026

Instructor.

Time and location.

4:35pm - 5:50pm on Mondays and Wednesdays in 427 Kaprielian Hall (KAP).

Registration.

Office hours.

Prerequisites.

We will assume familiarity the material from Math 540a and Math 535a.

About.

This is a second semester graduate level course in algebraic topology. We will cover various advanced topics, sampling many of the highlights from the 20th century while building the foundations for modern research in topology.

Possible topics.

• Poincaré duality
• generalized cohomology theories
• stable homotopy theory
• vector bundles
• Eilenberg-MacLane spaces
• equivariant cohomology
• characteristic classes
• cohomology operations
• topological K theory
• Bott periodicity
• spectral sequences

Textbooks and references.

• Characteristic Classes - Milnor and Stasheff [MS]
• Vector Bundles and K-Theory - Hatcher [H1]
• K-Theory - Atiyah [A]
• Cohomology Operations and Applications in Homotopy Theory - Mosher and Tangora [MT]
• A User's Guide to Spectral Sequences - McCleary [Mc]
• An Introduction to Homological Algebra - Weibel [W]
• Spectral Sequences in Algebraic Topology - Hatcher [H2]
• You Could Have Invented Spectral Sequences - Chow [C]

Problem sets.

Final presentations.

Grading scheme.

Tentative schedule